Prove that if $p(x)=\alpha x^2+ \beta x + \gamma$ is an interpolating polynomial of second degree of function $f$ (so $p(a)=f(a), p(b)=f(b), p(c)=f(c)$), passing through points $a<c<b$, then $$ \alpha = \frac{(b-c)f(a)+(c-a)f(b)+(a-b)f(c)}{(b-c)(a-c)(a-b)} \\ \beta = - \left(\frac{(b^2-c^2)f(a)+(c^2-a^2)f(b)+(a^2-b^2)f(c)}{(b-c)(a-c)(a-b)}\right) $$
I guess it should come from a set of equations: $$ p(a) = f(a)=\alpha a^2 + \beta a + \gamma\\ p(b) = f(b)=\alpha b^2 + \beta b + \gamma\\ p(c) = f(c)=\alpha c^2 + \beta c + \gamma $$ but I am completely clueless when solving it. I can't get neither $\alpha$ nor $\beta$ looking like in the thesis.
Could anyone give me a hand?
Expand
$$\alpha=\frac{\begin{vmatrix}f(a)&a&1\\f(b)&b&1\\f(c)&c&1\\\end{vmatrix}}{\begin{vmatrix}a^2&a&1\\b^2&b&1\\c^2&c&1\\\end{vmatrix}}$$
$$\beta=\frac{\begin{vmatrix}a^2&f(a)&1\\b^2&f(b)&1\\c^2&f(c)&1\\\end{vmatrix}}{\begin{vmatrix}a^2&a&1\\b^2&b&1\\c^2&c&1\\\end{vmatrix}}$$
The denominators are Vandermonde.